g)connections between and among multiple representations
of functions using verbal descriptions, tables, equations, and graphs;

h)
end behavior;

Adopted: 2016

BIG IDEAS

I will be able to model various kinds of mathematical relationships and express those relationships in different ways and I will be able to write symbolic representations of the way numbers behave and will know that in order to maintain equality, an operation performed on one side must also be performed on the other side.

UNDERSTANDING THE STANDARD

·Functions
describe the relationship between two variables where each input is paired to a
unique output.

·Function
families consist of a parent function and all transformations of the parent
function.

·The domain
of a function is the set of all possible values of the independent variable.

·The range of
a function is the set of all possible values of the dependent variable.

·For each x in the domain of f, x is a member of the
input of the function f, f(x) is a member of the output of f, and the ordered pair
(x, f(x)) is a member of f.

·A function
is said to be continuous on an interval if its graph has no jumps or holes in
that interval.

·The domain
of a function may be restricted algebraically, graphically, or by the practical
situation modeled by a function.

·Discontinuous
domains and ranges include those with removable (holes) and nonremovable
(asymptotes) discontinuities.

·A function
can be described on an interval as increasing, decreasing, or constant over a
specified interval or over the entire domain of the function.

·A function, f(x), is increasing over an interval if
the values of f(x) consistently
increase over the interval as the x
values increase.

·A function, f(x), is decreasing over an interval if
the values of f(x) consistently
decrease over the interval as the x
values increase.

·A function, f(x), is constant over an interval if
the values of f(x) remain constant
over the interval as the x values
increase.

·Solutions
and intervals may be expressed in different formats, including set notation,
using equations and inequalities, or interval notation. Examples may include:

Equation/Inequality

Set Notation

Interval Notation

x = 3

{3}

x = 3 or x = 5

{3, 5}

0 £ x £
3

{x|0 £ x £ 3}

[0, 3)

y ≥ 3

{y: y ≥ 3}

[3, ¥)

Empty
(null) set ∅

{ }

·A function, f, has an absolute maximum located at x = a
if f(a) is the largest value of f over its domain.

·A function, f, has an absolute minimum located at x = a
if f(a) is the smallest value of f over its domain.

·Relative
maximum points occur where the function changes from increasing to decreasing.

·A function, f, has a relative maximum located at x = a
over some interval of the domain if f(a)
is the largest value of f on the interval.

·Relative
minimum points occur where the function changes from decreasing to increasing.

·A function, f, has a relative minimum located at x = a
over some interval of the domain if f(a)
is the smallest value of f on the
interval.

·A value x in the domain of f is an x-intercept or a
zero of a function f if and only if f(x) = 0.

·Given a
polynomial function f(x), the
following statements are equivalent for any real number, k, such that f(k) = 0:

­k is a zero
of the polynomial function f(x) located
at (k, 0);

­k is a
solution or root of the polynomial equation f(x)
= 0;

­the point (k, 0) is an x-intercept
for the graph of f(x) = 0; and

­(x – k)
is a factor of f(x).

·Connections
between multiple representations (graphs, tables, and equations) of a function
can be made.

·End behavior
describes the values of a function as x
approaches positive or negative infinity.

·If (a, b)
is an element of a function, then (b,
a) is an element of the inverse of
the function.

·The reflection of a function over
the line represents the inverse of the reflected
function.

·A function
is invertible if its inverse is also a function. For an inverse of a function
to be a function, the domain of the function may need to be restricted.

·Exponential
and logarithmic functions are inverses of each other.

·Functions
can be combined using composition of functions.

· Two functions, f(x) and g(x), are
inverses of each other if f(g(x)) = g(f(x)) = x.

ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

· AII.7ade1Identify
the domain, range, zeros, and intercepts of a function presented algebraically
or graphically, including graphs with discontinuities.

·AII.7a1Describe
a function as continuous or discontinuous.

·AII.7c1Identify
the location and value of absolute maxima and absolute minima of a function
over the domain of the function graphically or by using a graphing utility.

·AII.7c2Identify
the location and value of relative maxima or relative minima of a function over
some interval of the domain graphically or by using a graphing utility.

·AII.7f1 For any x
value in the domain of f, determinef(x).

·AII.7g1Represent
relations and functions using verbal descriptions, tables, equations, and
graphs. Given one representation, represent
the relation in another form.

·AII.71Investigate
and analyze characteristics and
multiple representations of functions with a graphing utility.

·AII.7b1 Given the graph of a function, identify intervals on which the
function (linear, quadratic, absolute value, square root, cube root,
polynomial, exponential, and logarithmic) is increasing or decreasing.